It all depends on your role, your seniority and your posting. Some Movers (mobile teams on 1 AMW for example) are away a lot, but that's their role. Movers on main bases tend to deploy when their aircraft deploy - a few times a year. Again, depends on where, how long and how senior you are.
When they travel, it is usually to then manage the movement of equipment and personnel to and from where they are required. Again, some of these can be exercises, some operations. Some Movers based at Northolt are the teams that go to civilian airfields and move troops through as required.
Postings are postings. They move you to where you are needed. Look at where the RAF are. That is also where Movers are. The main Movers hubs are Brize Norton and Akrotiri, but there are Movers everywhere else (almost). There are Movers in Canada, USA, Africa, Europe and there are also Movers supporting specialist tasks (Special Forces etc).
If you want to travel, great. If you want to stay at home, don't join as a Mover.